The fact that a digital filter has a finite capacity and must therefore include a quantizer to adjust the internal word length used in the filter is the common cause for a variety of discrepancies between the actual filter behavior and that of the underlying ideal linear model which the filter is intended to imitate. One form of distortion that is of primary concern is the widely studied topic of limit cycles, which can have two forms. First, limit cycles called parasitic oscillations can occur when the filter is being driven, and these effects are extremely disadvantageous for steady state or period two inputs. Second, limit cycles called self-sustained oscillations can occur even when the filter is intended to be idle (i.e., not filtering) with a zero input.
The problem caused by limit cycles which arise due to nonlinearity of the quantization operation is compounded by the feedback inherent in the filter, and by the fact that several filter sections are usually cascaded. The severity of the problem can be appreciated by considering that for typical second order digital filters, the amplitudes of limit cycle oscillations can be as high as one to two orders of magnitude greater than the largest quantization or round-off error in a single iteration. For perspective, it is to be noted that in some applications the entire normal operating range of a typical filter is roughly three to four orders of magnitude greater than the single iteration quantization error.
Various different approaches have been suggested to deal with the problem of limit cycles. One common technique to simply to increase the internal word length of the filter beyond the encoding accuracy of the input signal. This approach allows limit cycles to exist, but eliminates their effect on the system in which the filter is employed. The expenses of such a solution can be prohibitive when it is realized that word length must be nearly doubled to completely eliminate limit cycle distortion. In another approach, which is designed to eliminate or at least reduce the amplitude of limit cycles, Buttner ("A Novel Approach to Eliminate Limit Cycles in Digital Filters with a Minimum Increase in the Quantization Noise", Proc. 1976 IEEE Intl. Symp. Circuits and Systems, pp. 291-294, April 1976) and Lawrence and Mina ("Control of Limit Cycles in Recursive Digital Filters Using Constrained Random Quantization", IEEE Trans. on Acoustics, Speech and Signal Processing, Vol. 26, No. 1, April 1978, pp. 127-134) teach the introduction of random noise into the quantizer input, in order to obtain a "random rounding" characteristic. Also, Fetterweis and Meerkotter ("Suppression of Parasitic Oscillations in Wave Digital Filters", IEEE Trans. CAS-22, 1975, pp. 668-673) and Meerkotter and Wegener ("A New Second-Order Digital Filter without Parasitic Oscillations", AEU, Electronics and Communications, Band 29, 1975, pp. 312-314) have proposed structural changes in the direct form configuration of conventional digital filters to achieve limit cycle control. In yet another proposal, Butterweck ("Suppression of Parasitic Oscillations in Second-Order Digital Filters by Means of a Controlled Rounding Arithmetic", AEU (Archive Electtvonik Ubertragungstechik), Electronics and Communications, Band 29, 1975, pp. 371-374) introduced the concept of controlled rounding, wherein a memory capability is used to determine how the quantizer output should be treated.
While the above proposals produce acceptable results under certain circumstances, the types of limit cycles which are of concern in many practical applications are not altogether satisfactorily treated, and the solutions can often be complicated and expensive. For example, the Butterweck apparatus has the property that all self-sustained oscillations (i.e., those with zero input) are eliminated except those with period up to two. Unfortunately, the amplitude of these remaining limit cycles can be quite high, particularly in high Q filters with high and low frequency poles. This severely limits the attractiveness of the Butterweck approach insofar as this extremely useful type of filter is concerned.
In view of the foregoing, it is the broad object of the instant invention to provide a controlled rounding technique that can suppress both self-sustained oscillations and parasitic oscillations for d.c. and period two inputs, in all direct form second order digital filters for which the underlying linear system is stable. This technique should take account of both quantization effects, and also the effects of overflow, thereby enabling use of a finite state machine (a filter of the type described) to perform functions theoretically intended for the ideal model. The foregoing objective is desirably achieved in filters which have high and low frequency poles, which filters are the ones of highest commercial interest.